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Let $\mu$ be a probability measure. Let $\mathcal A$ be a collection of measurable sets and $D(\mathcal A)$ be the minimal $\lambda$-system (Dynkin system) containing $\mathcal A$.

Is $\mu(D)$ for $D\in\mathcal D(\mathcal A)$ determined by $\mu(A)$ for all $A\in \mathcal A$?

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Yes, and this is a big part of why Dynkin systems are of interest.

$\pi$-systems, on the other hand, have a simpler definition, but $\mu(A\cap B)$ is not determined by $\mu(A)$ and $\mu(B)$.

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